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Notation Let $S_{\bullet }$ be a simplicial set. It follows immediately from the definition that if there exists a category $\operatorname{\mathcal{C}}$ and a map $u: S_{\bullet } \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ which exhibits $\operatorname{\mathcal{C}}$ as a homotopy category of $S_{\bullet }$, then the category $\operatorname{\mathcal{C}}$ is unique up to isomorphism and depends functorially on $S_{\bullet }$. To emphasize this dependence, we will refer to $\operatorname{\mathcal{C}}$ as the homotopy category of $S_{\bullet }$ and denote it by $\mathrm{h} \mathit{S}_{\bullet }$.